Ball bearings have been used for decades in a considerable number of devices comprising mechanisms moving in rotation. Their role is to limit the friction between two members, one of which, the rotor, is mobile in rotation relative to the other, the stator, which is fixed to the base, replacing the phenomenon of sliding between the two members by a phenomenon of rolling. A ball or tapered rolling bearing is usually composed of two coaxial rings comprising raceways between which a set of rolling elements (balls or rollers) is arranged the spacing of which is kept constant. The rest of this description will be limited to the case of ball bearings, even though the invention is not limited to this case.
In this way, friction between the rotating members is considerably reduced, consequently reducing the energy required to keep the rotor in rotation or the thermal heating between these members.
The bearings' rings and the balls are typically made of metal, usually of bearing steel. Steel rings and ceramic balls are also used for some applications. Full ceramic bearings also exist.
As an improvement, it is common to produce a rolling bearing by combining a pair of ball bearings arranged between the rotor and the stator to keep them perfectly coaxial, so as to create a pivot linkage.
By “shaft” is meant the part of the member in contact with the bearings' inner rings; this member is either the stator when the central element of the bearing is fixed relative to the base or the rotor when the central element of the bearing is rotating relative to the base. By “hub” is meant the portion of the member in contact with the bearings' outer rings; this member is either, inversely to the shaft, the rotor or the stator. In all cases, the hub wraps around the shaft.
Such rolling bearings are typically used in mechanisms such as driving devices, machine tool spindles, turbines, pointing mechanisms, precision machinery, momentum wheels, such a momentum wheel being for example used in a satellite attitude control application.
During their assembly and operation, the various members making up a ball bearing are naturally subjected to deformation, resulting in the appearance of mechanical play between the members which may be detrimental either to the performance of the rolling bearings or to the precise pointing of the hub.
For precision mechanisms of the type mentioned above, ball bearings known as preloaded ball bearings are used to eliminate such play and to ensure good pointing accuracy. Preloading a pair of rolling bearings consists of applying permanently an axial tightening force on the bearings' sides. This force causes an elastic deformation between the raceways and balls and generates a contact pressure, which allows the play to be eliminated.
To control and optimize the friction, paired rolling bearings are often used, to which a preload is applied as shown by FIGS. 1a and 1b. 
In FIG. 1a, which illustrates a bearing that has not yet been preloaded, two ball bearings 1g, 1d are juxtaposed. Only half of the bearings are shown, the bearings' axis of symmetry Z being horizontal in the figure. The inner 2g, 2d and outer 3g, 3d rings are shown in cut view. In this example, the inner raceways 4g, 4d, and the outer raceways 5g, 5d define normals 6g, 6d in contact with the balls tilted at a predefined specific angle of contact α. The normals 6g and 6d converging towards the outside of the bearings, this is called a back-to-back assembly. Inverting the bearings 1g and 1d would lead to an assembly configuration called “face-to-face” in which the normals in contact would converge towards the inside of the paired bearing. The balls 7g, 7d are simply in contact with their respective internal and external raceways. On purpose, some play 8 remains between the two inner rings 2g, 2d, whereas the two outer rings 3g, 3d are in contact.
In the configuration of FIG. 1b, a preload is applied as an axial force Pr, tightening the inner rings 2g, 2d against each other. Here, this preload causes an elastic deformation 9g, 9d (greatly exaggerated here for the purposes of the figure) of each ball 7g, 7d and of the raceways, preventing the emergence of play in the assembly. When the play is removed by applying the preload, the oblique angle of contact α increases slightly (typically by a few percent). The resulting angle of contact is known and produced to within a few degrees by bearing manufacturers, typically in a range varying from 10 to 40 degrees.
There are many ways to achieve this preload. The preload is called “rigid” when it is obtained by imposing the motion of one ring relative to the other. Only the case of rigid preloads will be considered here.
The preload is thus an important property of the rolling bearing. It helps conferring to it a defined and controlled stiffness. It also has a direct influence on the allowable loading level and rotor speed. The challenge for the designer is to ensure a controlled and steady preload over time.
The following is an embodiment known from the state of the art that provides both a rigid preload and an adequate rocking stiffness of the bearing, i.e. around axes perpendicular to the bearing rotation axis.
A rolling bearing device, of the type comprising a central shaft and a hub mobile in rotation relative to each other, comprises at least two rolling bearings, one designated “lower” and the other designated “upper”, arranged between the central shaft and the hub in two positions spaced in the axial direction (i.e. by definition along the Z axis). These rolling bearings comprise inner and outer rings and balls; the inner ring of each bearing is adjusted around the central shaft and the outer ring is adjusted inside the hub; the bearings are mounted back-to-back or face-to-face. A rigid preload is applied to these rolling bearings in the axial direction. Spacers are placed between the rolling bearings; with a spacer designated “inner” resting, by its upper and lower ends, on the inner rings of the upper and lower rolling bearings, respectively; a spacer designated “outer” resting, by its upper and lower ends, on the outer rings of the upper and lower rolling bearings, respectively.
A rigid preload is applied to the bearings in the axial direction using assembly elements designed for this purpose. For example, mounting flanges are used to tighten the bearings' rings onto the spacers.
This type of mounting is advantageous because it allows the bearings to be spaced out at an adjustable distance depending on the length of the spacers. According to the state of the art, the length of the spacers is chosen to be sufficiently large to provide with a high pointing accuracy and with the required rocking stiffness. The longer the spacers are, the greater the bearing's rocking stiffness will be. However, according to this state of the art, the spacers will have the minimum length allowing the required stiffness to be achieved in order to minimize the size and weight of the bearing.
This device, which is advantageous in terms of rocking stiffness, bulk and weight, has nevertheless a significant limitation for some applications that require the preload to remain substantially constant during temperature changes.
It is clear indeed that the preload is substantially altered when a ball bearing is subjected to a variation of temperature. It may increase, which degrades the friction torque and the life duration of the members in contact owing to wear. It may also decrease to the point where the preload is completely lost, in which case the resulting mechanical backlash degrades the pointing accuracy as well as the life duration, due to shocks generated within the bearing.
Two types of temperature variations are considered here: excursions and gradients:                temperature excursion refers to a uniform temperature variation, with a rolling bearing homogeneous in temperature, this one varying over time;        temperature gradient refers to a temperature variation in space, from one end to the other of the bearing. A radial gradient (where the shaft is warmer or colder than the hub) alters the preload significantly. In contrast, an axial gradient (one rolling bearing is warmer than the other) has little effect on the preload.        
A temperature excursion occurs frequently during operation of the system, for example because the bearings heat up especially at high rotational speeds or because of the presence of dissipative elements operating in close proximity, such as electronics, for example.
A temperature gradient is commonly occurring because the stator has a high thermal coupling with the base, whereas the thermal conduction from the rotor to the base passes through the bearings' balls that only provide a reduced thermal path, especially when they are made of low thermal conductivity materials, in particular in the case of ceramic balls.
In order to limit the preload variation in the case of a temperature excursion, it is known to use exclusively for all the parts of the bearing (including the balls) only materials that have the same thermal dilatation coefficient (also known as coefficient of thermal expansion or coefficient of thermoelastic expansion), for example steel. This yields a preload that remains constant when the bearing is subjected to a temperature excursion, because the excursion generates an isotropic expansion of the bearing, the axial and radial expansions being proportional, with a contact angle that remains constant.
In contrast, even in this case of materials with the same coefficient of expansion, when the bearing is subjected to a radial temperature gradient, the expansion of the bearing is no longer isotropic and the axial and radial expansions are only partially offset, with an oblique angle of contact that varies significantly. The preload can be affected significantly by this.